Minimization of the Principal Eigenvalue for an Elliptic Boundary Value Problem with Indefinite Weight (Viscosity Solutions of Differential Equations and Related Topics)

Minimization

of

the

Principal Eigenvalue for

an

Elliptic Boundary

Value

Problem

with

Indefinite

Weight

東北大学・理・数学

柳田英二

Eiji Yanagida

Mathematical Institute, Tohoku University

Sendai 980-8578, Japan

This article is based

on a

joint work with Chiu-Yen Kao and Yuan Lou of the Ohio State University. We consider the eigenvalue problem

(EVP) $\{\begin{array}{ll}\triangle\phi+\lambda m(x)\phi=0 in \Omega,\frac{\partial\phi}{\partial n}=0 on \partial\Omega,\end{array}$

where $\Omega$ is a bounded domain in $\mathbb{R}^{N},$ $m(x)$ is

an

indefinite weight. If the

eigenvalue problem (EVP) has a positive eigenfunction $\phi\in H^{1}(\Omega)$, then $\lambda$

is called a principal eigenvalue. Clearly, $\lambda=0$ is a principal eigenvalue with an associated eigenfunction $\phi\equiv 1$.

Here, we explain biological background of the above problem. Let us

consider the logistic equation

where $t$ is the time, $u(t)$ is the population of

some

biological species, $\lambda>0$

is a selection pressure, $m$ is an intrinsic growth rate. As is easily seen, if

$m>0$ then $u(t)arrow m$ as $tarrow\infty$ so that the biological species can survive.

Conversely, if $m\leq 0$, then $u(t)arrow 0$ as $tarrow\infty$ so that the species becomes

extinct.

Now, let us introduce a spatial variable $x$, and consider the difFusive logistic model (or Fisher-KPP equation)

(F) $\{\begin{array}{ll}\frac{\partial}{\partial t}u=\triangle u+\lambda u\{m(x)-u\} in \Omega,\frac{\partial u}{\partial n}=0 on \partial\Omega,u(x, 0)>0, in \overline{\Omega}.\end{array}$

In this model, $\Omega$ denotes the habitat and $u(x, t)$ represents the population

density.

When $m(x)$ changes its sign depending on $x$, then a region with $m(x)>0$ is favorable to the species whereas a region with $m(x)<0$ is unfavorable. In the

case

where $m(x)$ changes its sign,

can

the species

survive?

In order to

answer

this question, it suffices to consider the stability of the trivial steady state $u\equiv 0$. Indeed, it holds that

$u=0$ is unstable $\Leftrightarrow$ solutions leave away from $0\Leftrightarrow$ survival,

$u=0$ _{is stable $=$ solutions tend to} $0\Leftrightarrow$ extinction.

On the other hand, the stability of the trivial solution

can

be determined by analyzing the linearized eigenvalue problem

(LEP) $\{\begin{array}{ll}\mu\Phi=\triangle\Phi+\lambda m(x)\Phi in \Omega,\frac{\partial\Phi}{\partial n}=0 on \partial\Omega.\end{array}$

$\mu_{0}>0\Leftrightarrow$ unstable $\Leftrightarrow$ survival,

$\mu_{0}<0\Leftrightarrow$ stable $\Leftrightarrow$ extinction.

By the eigenvalue analysis, we can show the following about the maximal

eigenvalue of (LEP):

Case I: If $m(x)<0$

on

$\Omega$, then

$\mu_{0}<0$ for any $\lambda>0$. . . . extinction, Case II: If $\int_{\Omega}m(x)dx>0$, then $\mu>0$ for any $\lambda>0$. . . . survival.

Case

III: If $m(x)$ changes its sign and $\int_{\Omega}m(x)dx<0$, then the positive

principal eigenvalue $\lambda_{p}>0$ of (EVP) has the following properties:

(i) If $0<\lambda<\lambda_{p}$, then $\mu_{0}<0$. . . . extinction

(ii) If $\lambda>\lambda_{p}$, then _{$\mu_{0}>0$}. . . . survival.

For an endangered species, we control $m(x)$ by some

means

_{in order to}

reserve

thespecies. In this case, if the positive principal eigenvalue $\lambda_{p}$ issmall,

then the species has

a

better chance to survive. Under

a

limited resource,

how

can

we minimize $\lambda_{p}$? This question is formulated mathematically

as

follows. For the eigenvalue problem,

we

impose the following conditions:

(Al) $\Omega_{+}:=\{x\in\Omega : m(x)>0\}$ has positive

measure.

(A2) $\int_{\Omega}m(x)dx<0$.

(A3) For a fixed constant $\kappa>0,$ $m(x)$ satisfies $-1\leq m(x)\leq\kappa$ a.e. on $\Omega$.

(A4) For

a fixed

constant $0<\mu<1,$ $m(x)$ satisfies $\int_{\Omega}m(x)dx\leq(-1+\mu)|\Omega|$.

Let $\mathcal{M}$ denote the set of $m(x)$ satisfying the above conditions. In (A3),

$m(x)=-1$ corresponds to a growth ratewithout any protection, and $m(x)=$

$\kappa$ is a growth rate in the optimal environment. In (A4), the constant

$\mu$

corresponds to the maximal supply of the

resource

and the inequality $\mu<1$

implies that the resource is not sufficient.

Under the constraints (Al) and (A2), Brown-Lin [1] and Senn-Hess [7]

showed that (EVP) has

a

unique positive principal eigenvalue $\lambda_{p}(m)$.

Saut-Scheurer [6] proved that

$\lambda_{\inf}:=\inf_{m\in \mathcal{M}}\lambda_{p}(m)>0$.

Cantrell-Cosner

[2] addressed the following question: Among all $m(x)\in$

$\mathcal{M}$, which $m(x)$ minimizes

$\lambda_{p}(m)$? They studied

some

simple cases, but their answer was not satisfactory. In the following, we derive some general

properties for this problem, and determined the minimizer in

some

specific

cases. First, concerning the minimizer of $\lambda_{p}(m)$, the following bang-bang

property holds:

Theorem 1. ([5]) The

_{infimum of}

$\lambda_{\inf}$

of

$\lambda_{p}(m)$ is attained by

some

$m\in \mathcal{M}$,

and such $m$ is expressed as

follows

by using a measurable set $E\subset\Omega.\cdot$

$m(x)=\{\begin{array}{ll}\kappa for x\in E,-1 for x\not\in E,\end{array}$ _$a.e$.

Proof First, by the variational principle, the positive principal eigenvalue of

(EVP) is characterized by using the Rayleigh quotient

$\lambda_{p}(m)=$ $\inf$

$\underline{\int_{\Omega}|\nabla U|^{2}}$

$U \in S(m)\int_{\Omega}m(x)U^{2}$

where

$S(m):=\{U\in H^{1}(\Omega)$ : $\int_{\Omega}m(x)U^{2}>0\}$ .

Moreover, $\lambda_{p}(m)$ is simple, and the infimum of the Rayleigh quotient is at-tained only by associated eigenfunctions that do not change sign. Suppose

now that $m_{1}(x)$ is not of bang-bang type, and let $phi_{1}$ be a positive

eigen-function associated with the principal eigenvalue $\lambda_{p}(m_{1})$. Let $m_{2}(x)$ be a

weight function obtained by moving

resource

from

a

region with smaller $\phi_{1}$

to that with larger $\phi_{1}$, and let $\phi_{2}$ be an positive eigenfunction associated with

the principal eigenvalue $\lambda_{p}(m_{2})$. Then we have

$\lambda_{p}(m_{1})=\frac{\int_{\Omega}|\nabla\phi_{1}|^{2}}{\int_{\Omega}m_{1}(x)\phi_{1}^{2}}>\frac{\int_{\Omega}|\nabla\phi_{1}|^{2}}{\int_{\Omega}m_{2}(x)\phi_{1}^{2}}\geq\frac{\int_{\Omega}|\nabla\phi_{2}|^{2}}{\int_{\Omega}m_{2}(x)\phi_{2}^{2}}=\lambda_{p}(m_{2})$ .

Therefore, if $m(x)$ is not of bang-bang type, the it cannot be a minimizer.

(We omit details for the existence of

a

minimizer.) $\square$

By Theorem 1, it suffices to determine the set $E\subset\Omega$ to fine a global

minimizer. Let us consider the one-dimensional problem

as

the simplest

case

(EVPI) $\{\begin{array}{ll}\phi_{xx}+\lambda m(x)\phi=0, x\in(0,1),\phi_{x}(0)=\phi_{x}(1)=0. \end{array}$

In this case, the constraints are described as

$-1\leq m(x)\leq\kappa$, $-1< \int_{0}1_{m(x)dx}\leq-1+\mu<0$.

Theorem 2. ([5]) In the eigenvalue problem $(EVPl),$ $\lambda_{p}(m)=\lambda_{\inf}$ holds

if

and only

_if

$or$

$m(x)=\{\begin{array}{ll}-1 for x\in(0,1-\alpha),\kappa for x\in(1-\alpha, 1),\end{array}$ _$a.e.$, where $0<\alpha<1$ _{is chosen to be}

$\int_{0}1_{m(x)dx=-1+\mu}$.

Proof.

Given

a

weight function$m_{1}(x)$ and its associated eigenfunction $\phi_{1}(x)$,

we define its spatial rearrangement by

$m_{2}(\xi(s))=s$, $\xi(s)=$

measure

$\{x:m_{1}(x)>s\}$, $U(\eta(s))=s$, $\eta(s)=$

measure

$\{x : \phi_{1}(x)>s\}$.

Then we have

$|\nabla U|\leq|\nabla\phi_{1}|$, $\int_{\Omega}m_{1}(x)\phi_{1}^{2}\leq\int_{\Omega}m_{2}(x)U^{2}$.

Hence

$\lambda_{p}(m_{1})=\frac{\int_{\Omega}|\nabla\phi_{1}|^{2}}{\int_{\Omega}m_{1}(x)\phi_{1}^{2}}>\frac{\int_{\Omega}|\nabla U|^{2}}{\int_{\Omega}m_{2}(x)U^{2}}>\lambda_{p}(m_{2})$ .

口

By Theorem 2, in the Fisher-KPP model, the biological species has the

maximal chance to survive if the weight is at

one

end of the interval.

Corolllary 1. Let $\Omega=(0,1)$.

If

$\lambda\leq\lambda_{\inf}$, then

for

any $m\in \mathcal{M}$, the trivial solution

_of

Fisher-KPP model is globally stable.

Next, we consider a limiting problem of thin cylindrical domains:

(EVP2) $\{\begin{array}{ll}\frac{1}{a(x)}\{a(x)\phi_{x}\}_{x}+\lambda m(x)\phi=0, x\in(O, 1),\phi_{x}(0)=\phi_{x}(1)=0, \end{array}$

where $a(x)$ is a positive smooth function representing the width of the thin

domain. Without loss of generality, we may assume

$\int_{0}1_{a(x)dx=1}$

and

assume

also that

$-1\leq m(x)\leq\kappa$, $-1< \int_{0}^{1}m(x)a(x)dx\leq-1+\mu<0$.

Theorem 3. ([4]) For the eigenvalue problem $(EVP2)$, both

$m(x)=\{\begin{array}{ll}\kappa for x\in(0, \alpha),-1 for x\in(\alpha, 1),\end{array}$ _$a.e$.

and

$m(x)=\{\begin{array}{ll}-1 for x\in(0,1-\beta),\kappa for x\in(1-\beta, 1),\end{array}$ _$a.e$.

are

local minimizers, where $0<\alpha,$ $\beta<1$

are

constants such that $\int_{0}^{1}m(x)a(x)dx=-1+\mu$.

The proof is obtained by showing that the principal eigenvalue becomes

smaller if we perturb a positive region through the Rayleigh quotient. We note that the variational principle for (EVP2) is formulated

as

where

$S(m)$ $:=\{U\in H^{1}(\Omega)$ : $\int_{0}^{1}a(x)m(x)U^{2}>0\}$ .

We note _{that this result does not necessarily imply that there}

_are

_{other local} minimizers. In fact, there is an example of $a(x)$ for which another local

minimizer exists.

If the thin domain is not so constricted, then one of the local minimizers

obtained in Theorem

3

becomes a global minimizer.

Theorem 4. ([4]) For the eigenvalue problem (EVP2), there exists an

$R=R(\kappa, \mu)>1$ such that if

$\frac{\max a(x)}{\min a(x)}<R$,

then

$m(x)=\{\begin{array}{ll}\kappa for x\in(0, \alpha),-1 for x\in(\alpha, 1),\end{array}$ _$a.e$.

or

$m(x)=\{\begin{array}{ll}-1 for x\in(0,1-\beta),\kappa for x\in(1-\beta, 1),\end{array}$ _$a.e$.

is a global minimizer.

The value of the constant $R$ is important, but its best constant is not

clear.

Next, let us consider the case where $\Omega$ be

a

rectangular domain

$\Omega=\{(x, y)\in(0,1)\cross(0, a)\}\subset \mathbb{R}^{2}$

and that $m(x, y)$ is positive on a strip-like domain given by $E=(0, c)\cross(0, a)$_.

Theorem 5. ([3]) Fixing other parameters, there exists a critical value

Proof. We perturb the strip-like region $E$ as follows by using a small

pa-rameter $\epsilon>0$:

$E_{\epsilon}=(0, c+\epsilon\cos(j\pi/a))\cross(0, a)$, $j=1,2,$ _$\ldots$

Expanding $\lambda_{\epsilon}$ and the associated eigenfunction $\phi_{\epsilon}$

as

$\lambda_{\epsilon}=\lambda_{0}+\epsilon^{2}\lambda_{2}+o(\epsilon^{2})$,

$\phi_{\epsilon}=\phi_{0}+\epsilon^{2}\phi_{2}+o(\epsilon^{2})$,

substituting these expansions in the equation, and comparing termwise, then

we

can

determine the sign of $\lambda_{2}$ as

$c<\exists_{C^{*}}\Leftrightarrow\lambda_{2}<0\Rightarrow$ not a local minimizer.

Based on this formal analysis,

we can

obtain

a

rigorous proof by using the

Rayleigh quotient. 口

Formal argument suggests that if $c>c^{*}$, then the strip-like pattern is a

local minimizer, but it is difficult to verify it rigorously.

Theorem 5 implies that if the

resource

is not enough, then the strip-like

patternis not a local minimizer. By numerical computation with a projection

gradient method,

a

strip-like pattern in another side

or a

disc pattern

seems

to be a global minimizer. Here the idea of the projection gradient method is

to start from an initial guess for $m(x)$, evolve it along the gradient direction

until it reaches the optimal configuration. Since the gradient direction may

result in the violation of the constraint, a projection approach is used to project $m(x)$ back to the feasible set. Furthermore, we propose _a new _binary

update for $m(x)$ to preserve the bang-bang structure. See _{[3] for} more

Theorem 6. ([4]) In a rectangular domain, any global minimizer and its associated eigenfunction are both monotone in x-direction and y-direction.

The proof is based

on

spatial rearrangement. In fact, rearrangement in

both x-direction and y-direction make the principal eigenvalue smaller. The minimization problem in

more

general domains is an extremely

dif-ficult question. If we consider a singular limit of the problem, it may be

possible to characterize the minimizer. Even if

we

cannot characterize

a

global minimizer, it is desired to make clear its properties. It is conjectured that if the domain is convex, then the set $E$ is simply connected, but it is

still open.

References

[1] K. J. Brown and

S.

S. Lin,

On

the existence

_of

positive eigenvalue problem

with

_indefinite

weightfunction, J. Math. Anal. Appl. 75 (1980),

112-120.

[2] R. S. Cantrell and C. Cosner,

_Diffusive

logistic equations with

_indefinite

weights: population models in a disrupted environments, Proc. Roy. Soc.

Edinburgh $112A$ (1989),

293-318.

[3] C.-Y. Kao, Y. Lou and E. Yanagida, Principal eigenvalue for

an

ellip-tic problem with indefinite weight

on

cylindrical domains, Mathematical

Biosciences and Engineering 5 (2008),

315-335.

[4] C.-Y. Kao, Y. Lou and E. Yanagida, work in progress.

[5] Y. Lou and E. Yanagida, Minimization

_of

the principal eigenvalue

_for

an

elliptic boundary value problem with

_indefinite

weight and applications to

[6] J. C. Saut and B. Scheurer, Remarks on a nonlinear equation arising in

population genetics, Comm. Part. Diff. Eq. 23 (1978), pp. 907-931.

[7]

S. Senn

and P. Hess,

On

positive solutions

of

a

linear elliptic

bound-ary value problem with Neumann boundary conditions, Math. Ann. 258